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Proportionality (mathematics)
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, two quantities
Quantity

Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with Quality , substance, change, and relation....
 are called proportional if they vary in such a way that one of the quantities is a constant multiple
Multiple

The word multiple can refer to:*Multiple of numbers.*List of independent discoveries, instances of scientists, working independently of each other, reaching similar findings....
 of the other, or equivalently if they have a constant ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
.

Proportion also refers to the equality of two ratios.

n two variable
Variable

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e....
s x and y, y is (directly) proportional to x (x and y vary directly, or x and y are in direct variation) if there is a non-zero constant k such that

The relation is often denoted

and the constant ratio

is called the proportionality constant or constant of proportionality.








e

is equivalent to

it follows that if y is proportional to x, with (nonzero) proportionality constant k, then x is also proportional to y with proportionality constant 1/k.

If y is proportional to x, then the graph of y as a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 of x will be a straight line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 passing through the origin
Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special Point , usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space....
 with the slope
Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
 of the line equal to the constant of proportionality: it corresponds to linear growth.

oted in the definition above, two proportional variables are sometimes said to be directly proportional.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, two quantities
Quantity

Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with Quality , substance, change, and relation....
 are called proportional if they vary in such a way that one of the quantities is a constant multiple
Multiple

The word multiple can refer to:*Multiple of numbers.*List of independent discoveries, instances of scientists, working independently of each other, reaching similar findings....
 of the other, or equivalently if they have a constant ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
.

Proportion also refers to the equality of two ratios.

Direct proportion

Given two variable
Variable

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e....
s x and y, y is (directly) proportional to x (x and y vary directly, or x and y are in direct variation) if there is a non-zero constant k such that

The relation is often denoted

and the constant ratio

is called the proportionality constant or constant of proportionality.

Examples

  • If an object travels at a constant speed
    Speed

    Speed is the rate of Motion , or equivalently the rate of change of distance.Speed is a Scalar quantity with dimensions length/time; the equivalent Vector quantity to speed is velocity....
    , then the distance
    Distance

    Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
     traveled is proportional to the time
    Time

    Time is a component of the measurement used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects....
     spent travelling, with the speed being the constant of proportionality.


  • The circumference
    Circumference

    The circumference is the distance around a closed curve. Circumference is a kind of perimeter....
     of a circle
    Circle

    A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
     is proportional to its diameter
    Diameter

    In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle....
    , with the constant of proportionality equal to p
    Pi

    Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
    .


  • On a map
    Map

    A map is a visual representation of an area?a symbolic depiction highlighting relationships between elements of that space such as Object , regions, and topic-comment....
     drawn to scale
    Scale (map)

    Sorry, no overview for this topic
    , the distance between any two points on the map is proportional to the distance between the two locations the points represent, with the constant of proportionality being the scale of the map.


  • The amount of force acting on a certain object (its weight
    Weight

    In the physical sciences, weight is a measurement of the gravitational force acting on an object. Near the surface of the Earth, the Earth's gravity is approximately constant; this means that an object's weight is roughly proportional to its mass....
    ) from the gravity of the Earth
    Earth

    Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
     at sea level is proportional to the object's mass
    Mass

    In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
    , with the gravitational acceleration
    Gravitational acceleration

    In physics, gravitational acceleration is the acceleration of an object caused by the force of gravity from another object. In the absence of any other forces, any object will accelerate in a gravitational field at the same rate, regardless of the mass of the object....
     being the constant of proportionality on the object.


Properties

Since

is equivalent to

it follows that if y is proportional to x, with (nonzero) proportionality constant k, then x is also proportional to y with proportionality constant 1/k.

If y is proportional to x, then the graph of y as a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 of x will be a straight line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 passing through the origin
Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special Point , usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space....
 with the slope
Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
 of the line equal to the constant of proportionality: it corresponds to linear growth.

Inverse proportionality

As noted in the definition above, two proportional variables are sometimes said to be directly proportional. This is done so as to contrast proportionality with inverse proportionality.

Two variables are inversely proportional (or varying inversely, or in inverse variation) if one of the variables is directly proportional with the multiplicative inverse
Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1....
 of the other, or equivalently if their product
Product (mathematics)

In the a mathematics, a product is the result of Multiplication, or an expression that identifies divisors to be multiplied. The order in real number or complex number numbers are multiplied has no bearing on the product; this is known as the Commutativity of multiplication....
 is a constant. It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that

The constant can be found by multiplying the original x variable and the original y variable.

Basically, the concept of inverse proportion means that as the absolute value
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
 or magnitude of one variable gets bigger, the absolute value or magnitude of another gets smaller, such that their product (the constant of proportionality) is always the same.

For example, the time taken for a journey is inversely proportional to the speed of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging.

The graph of two dancing variables varying inversely on the Cartesian coordinate plane is a hyperbola
Hyperbola

In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
. The product of the X and Y values of each point on the curve will equal the constant of proportionality (k). Since k can never equal zero, the graph will never cross either axis.

Hyperbolic coordinates

The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates
Hyperbolic coordinates

In mathematics, hyperbolic coordinates are a useful method of locating points in Quadrant I of the Cartesian plane in takeand.Sometimes the parameter is called hyperbolic angle and the geometric mean....
; the two coordinates correspond to the constant of direct proportionality that locates a point on a ray
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 and the constant of inverse proportionality that locates a point on a hyperbola.

Exponential and logarithmic proportionality

A variable y is exponentially proportional to a variable x, if y is directly proportional to the exponential function
Exponential function

The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
 of x, that is if there exists a non-zero constant k

Likewise, a variable y is logarithmically proportional to a variable x, if y is directly proportional to the logarithm
Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
 of x, that is if there exists a non-zero constant k

Experimental determination

To determine experimentally whether two physical
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
 quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
. If the points lie on or close to a straight line that passes through the origin (0, 0), then the two variables are probably proportional, with the proportionality constant given by the line's slope
Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
.

See also

  • Correlation
    Correlation

    In probability theory and statistics, correlation indicates the strength and direction of a linear relationship between two random variables....
  • Eudoxus of Cnidus
    Eudoxus of Cnidus

    Eudoxus of Cnidus was a Ancient Greece astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy....
  • Golden ratio
    Golden ratio

    In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
  • Proportional font
  • Rule of three (mathematics)
  • Sample size
    Sample size

    The sample size of a statistical sample is the number of observations that constitute it. It is typically denoted n, a positive integer ....
  • Similarity
    Similarity

    Similarity is some degree of symmetry in either analogy and resemblance between two or more concepts or physical objects. The notion of similarity rests either on exact or approximate repetitions of patterns in the comparison items....

Growth

  • Linear growth
  • Hyperbolic growth
    Hyperbolic growth

    When a quantity grows towards a Mathematical singularity under a finite variation it is said to undergo hyperbolic growth.More precisely, the reciprocal function has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as is infinity: any similar graph is said to exhibit hyperbolic growth....